Question

in the figure, m∠dbc = m∠cad, m∠dba = m∠cab, and ec = ed. if m∠cba = 5x + 2, ec = 3, and eb = 8, find each value. see examples 3 and 5

1. m∠dbc 30. m∠cba 31. m∠cab
2. bd 33. m∠dba 34. m∠dab
3. points p, q, r, and s are collinear. point q is between p and r, r is between q and s, and pq = rs. if ps = 18 and pr = 15, what is the value of qr?

lesson 1 - 1 measuring segments and angles 11

Explanation:

Step1: Set up equation for angles

Since \(m\angle DBA=m\angle CAB\), we have \(3x - 8=2x + 1\).
\[3x-2x=1 + 8\]
\[x = 9\]

Step2: Find \(m\angle DBC\)

We know \(m\angle DBC=m\angle CAD\) and from angle - relationships. First, find \(m\angle CBA=5x + 2\). Substitute \(x = 9\) into \(m\angle CBA\): \(m\angle CBA=5\times9+2=45 + 2=47\). Also, since \(m\angle DBA=3x - 8\) and \(x = 9\), \(m\angle DBA=3\times9-8=27 - 8 = 19\). Then \(m\angle DBC=m\angle CBA - m\angle DBA=47-19 = 28\).

Step3: Find \(m\angle CBA\)

Substitute \(x = 9\) into \(m\angle CBA=5x + 2\). So \(m\angle CBA=5\times9+2=47\).

Step4: Find \(m\angle CAB\)

Substitute \(x = 9\) into \(m\angle CAB=2x + 1\). So \(m\angle CAB=2\times9+1=19\).

Step5: Find \(BD\)

Since \(EC = ED = 3\) and \(EB = 8\), and using triangle - related properties (not enough information given to fully justify a non - geometric - guess method, assume some congruence or similarity which is not explicitly stated, but if we consider the figure in a general triangle - context), we cannot find \(BD\) with the given information.

Step6: Find \(m\angle DBA\)

Substitute \(x = 9\) into \(m\angle DBA=3x - 8\), so \(m\angle DBA=19\).

Step7: Find \(m\angle DAB\)

\(m\angle DAB=m\angle DBA+m\angle CAB\) (angle addition in \(\triangle DAB\)), so \(m\angle DAB=19 + 19=38\).

Step8: Solve for \(QR\) in collinear points problem

We know \(PS=PQ + QR+RS\) and \(PQ = RS\), \(PS = 18\), \(PR=PQ + QR = 15\). Let \(PQ = RS=y\) and \(QR = z\). Then \(2y+z=18\) and \(y + z=15\). Subtract the second equation from the first: \((2y + z)-(y + z)=18 - 15\), \(y = 3\). Substitute \(y = 3\) into \(y + z=15\), we get \(z=12\), so \(QR = 12\).

Answer:

1. \(m\angle DBC = 28\)
2. \(m\angle CBA = 47\)
3. \(m\angle CAB = 19\)
4. Cannot be determined
5. \(m\angle DBA = 19\)
6. \(m\angle DAB = 38\)
7. \(QR = 12\)